Depending on speeds depends on cables. If you think you might in the future make extensive use on 10Gbps, use CAT6, but that has lower cable length limits, and is also more difficult to terminate well. CAT5 is good enough for 1Gbps, but in reality is limited to that, although you can often get more than that - the socket in my kitchen is normal phone grade 6 wire (not rated even to 10Mbps) and I'm running it at 100Mbps over about a 7m run.
Be aware of the lower cost CAT5E cables about, they are not solid copper, but something called Copper Clad Aluminium (CCA) - which despite the Cat5e naming the manufacturers may give it is NOT Cat5e compliant. It would also appear to degrade over time.
Personally, I'd go CAT6, for future proofing. And make sure you run enough everywhere. When I moved here 17.5yrs ago, I never envisaged needing networking behind the TV. Now I have to keep swapping cables over behind there as I only have 2, and the TV is permanently on wifi
(and only does 2.4Ghz, which is a massive problem where I live)
If running and terminating yourself, there are plenty of utube vids and web tutorials. Watch a few, as some are BS. If you've never done it before, you'll find it fiddly, but just keep practicing and practicing. Also, be aware there are 2 different common wiring standards, 568A and 568B. Ensure *everything* is wired to same standard - some faceplates might be colour codes to 586A, some 568B, but most nowadays are marked with both (in itself could be confusing
).
Any cable going outdoors should really be shielded twisted pair network cable, with each end earthed. These are more difficult to run, and putting the connectors on are a bit of a bitch
Depending on your needs, I'd probably fit a reasonably decent switch in house and garage, probably 1Gbps for now, but once 10Gbps stuff drops, leave yourself the option to swap switches out in 2 or 3 years, and have a 10Gbps link between the 2... ...obviously depending on how much data your will be pushing between the 2 buildings (eg, if you ever think you'll have a central storage device, you will want ultra fast access to it). Also, decide if you are going to need PoE on the switches. The ex-Linksys Cisco Small Business switches are worth a good look (SG200 series for layer 2 only, or SG300 or SG500 for layer 3 - all can do vlan, qos etc), and reasonably priced... ...but very different beasts from the full managed Cisco stuff. If you don't mind rebooting the switches regularly due to ports not waking up, the non-PoE Netgear GS724/GS748 are very good value, and completely silent due to fanless designs.
For garage wifi access point, Ubiquiti do some bloody good APs now. Bit of a pig to set up sometimes, but are getting better, and I think you'd have no problem setting it up. Defo worth a look. Nowadays, go for a minimum of AC spec on the 5Ghz band. Everything will be >300Mbps on 2.4Ghz band. (5Ghz has better speed, but doesn't penetrate walls well. 2.4Ghz has less speed, more contention (with Bluetooth, video senders, other wifi, microwaves etc etc) but better penetration.
For routers, drop me a PM - what I use here is also an important part of security, so I tend not to broadcast it, as its the first link from the Internet.